Econ. 553a
Fall 2013
Peter C.B. Phillips
Econometrics IV: Time Series Econometrics
Course Outline 2013
This is a one semester version of what was originally a two-course sequence in time series econometrics that
included Econ 557b. The course provides an introduction to time series methods in econometrics covering
stationary series, aspects of trend behavior, detrending mechanisms and their properties, unit root theory,
cointegrated system approaches, realized volatility and quarticity, Wold and BN decompositions, model
selection, nonlinear nonstationary models and methods, spatial density asymptotics, and long memory
modeling. Both time domain and frequency domain methods are discussed, and Bayesian as well as
classical approaches are included. The treatment relies on asymptotic theory for linear processes,
martingales and martingale approximations. We overview a large literature and not all topics are treated in
the same depth. Theory, computations and some empirical applications are discussed. Most classes are
divided into two parts, one dealing with theory and the other with empirics.
No specific text is recommended. The most recent text is Martin, Hurn and Harris (2012), which provides
good general coverage at the introductory level in an approach that is oriented towards implementation
offering abundant illustrations (uniquely) complete with computer code in Gauss, Matlab and R.
Hamilton’s (1994)1 book, Fuller (1996) and Gourieroux and Monfort (1997) are useful references.
Hamilton’s coverage is broad and relevant to econometrics, the book is easy to read and it includes much
introductory material, but is now dated. Fuller’s book provides an accessible statistical treatment of the
subject, is a useful revision of an earlier (1976) edition, and was the first text to discuss unit root theory.
Gourieroux and Monfort (1997) is a translation of an excellent French textbook of time series that covers a
wide literature from an econometric perspective. Lutkepohl and Kratzig (2004) is a textbook of applied
time series econometrics that emphasizes practicalities and covers methods that are popular in empirical
economic applications. Brockwell and Davis (1991, with subsequent editions) is a very successful time
series text that is commonly used in North American graduate statistics courses. This book is more technical
than the above texts and stresses univariate models, but is well exposited, covers most of the traditional
stationary time series topics and comes with some computer software. Lutkepohl’s (1993) book and his
newer (2005) text provide excellent coverage and exposition of VAR and Bayesian VAR modelling
methods, together with some small scale practical applications to macro data. Hall and Heyde (1980) is a
beautifully written classic on martingale limit theory that continues to reward careful reading. Billingsley
(1999) is the second edition of a highly influential treatise on weak convergence that first appeared in 1968.
Davidson (1994) is a good general reference source on limit theory for econometrics including functional
laws, emphasizing mixing and weak dependence. Van de Vaart (1998) is a useful overview of asymptotic
methods in statistics, including some empirical process methods. Taniguchi and Kakizawa (2000) give a
modern treatment of time series asymptotics from a stochastic process perspective and include some useful
special topics like large deviation expansions, saddlepoint approximations and higher order asymptotics.
White (2002) provides much useful background and its first edition (1984) was notable for its general
treatment of asymptotic covariance matrix estimation.
A take home examination will be given at the end of the course. Students have the option of attempting a
solution to the problems in this exam, writing a scientific overview of a modern research area in
1
See Section 0 in the Reading Guide below for general references.
2
econometrics, or doing an applied econometrics paper on a topic of their choice. The empirical paper may
be used for the applied econometrics paper requirement. Past take home exams over many years are all
available on the web and some solution sets are available.
The following is a general outline of how we proceed through the course material. Some of the material is
taught in section based on earlier years’ lectures to help us cover more material in the course. We adjust
lecture content according to the rate of progress, importance of the material, and relevance to applied work.
Some empirical applications will be discussed.
Week
Content
1&2
Ideas and approaches to time series. Primary concerns and methods of inference:
Classical, Bayesian and prequential approaches. Role of unit roots and
cointegration in econometric modeling. Brownian motion, the Karhunen-Loève
representation, and some of its recent applications.
3&4
Heuristic ideas and implications for inference and modelling. Simple parametric
models, including VARs, and Cointegrated systems. Optimal estimation and IV
approaches. Some preliminary asymptotics. Information criteria and model
selection asymptotics. Trend Elimination.
5&6
Ergodicity and notions of weak dependence. Conditional expectations and
Hilbert projections. The Wold decomposition and forecasting. Grenander
Rosenblatt theory.
7&8
The Phillips-Solo device & shortcuts to time series asymptotics. Strong laws and
CLT’s for time series. Unit root limit theory, asymptotic degeneracy, realized
volatility, quadratic variation, and quarticity. Optimal estimation of cointegrated
systems.
9 & 10
Martingales and time series applications of the martingale convergence theorem.
Explosive and mildly explosive time series. Bubbles, crashes and applications in
finance.
11
Nonlinear nonstationary models, spatial density, estimation of local time.
Applications to market intervention.
12 & 13
Frequency domain approaches and spectral regression. Spectral density and long
run variance estimation. Long memory models and econometric methods. More
on unit roots and cointegration.
December - January
Take Home examination paper, overview paper, or applied econometrics paper
3
Reading Guide
Time series is a vast subject. The following list covers only that part of the subject that relates most closely
to econometric research. The list is subdivided into topics that are relevant to material we intend to discuss,
if only briefly in some cases, during the course.
0. General References2
Aoki, M. (1987) State Space Modeling of Time Series. New York: Springer.
Anderson, T.W. (1971) The Statistical Analysis of Time Series. New York: Wiley.
Banerjee, A., J. Dolado, J.W. Galbraith and D.F. Hendry (1993) Cointegration, Error-Correction and
the Econometric Analysis of Non-Stationary Data. Oxford: Oxford University Press.
Bierens, H. J. (1996) Topics in Advanced Econometrics: Estimation, testing and specification of cross
section time series models. Cambridge University Press.
* Billingsley, P. (1999) Weak Convergence of Probability Measures. Second Edition. New York:
Wiley.
Box, G.E.P. and G.M. Jenkins (1976) Time Series Analysis: Forecasting and Control, 2nd ed. San
Francisco: Holden Day.
* Brillinger, D.R. (1981) Time Series: Data Analysis and Theory, 2nd ed. San Francisco: Holden Day.
* Brockwell, P.J. and R.A. Davis (1986) Time Series: Theory and Methods. New York: Springer (2nd
ed., 1991).
Clements M. P. and D. F. Hendry (1998) Forecasting Economic Time Series. Cambridge: Cambridge
University Press.
* Davidson, J. (1995) Stochastic Limit Theory Oxford: Oxford University Press.
Dhrymes, P. (1989) Topics in Advanced Econometrics. New York: Springer Verlag.
Fan, J. and Q. Yao (2003) Nonlinear Time Series. Nonparametric and Parametric Methods. New
York: Springer.
* Fuller, W.A. (1996) Introduction to Statistical Time Series, 2nd Edition. New York: Wiley.
Fishman, G. (1969) Spectral Methods in Econometrics. Cambridge: Harvard University Press.
* Gourieroux C. and A. Monfort (1997). “Time Series and Dynamic Models”. Cambridge: Cambridge
University Press.
Granger, C.W.T. and P. Newbold (1987) Forecasting Economic Time Series, 2nd edition. New York:
Academic Press.
2
Asterisked references are more important to the course.
4
Grenander, U. and M. Rosenblatt (1957) Statistical Analysis of Stationary Time Series. New York:
Wiley.
* Hall, P. and C.C. Heyde (1980) Martingale Limit Theory and its Applications. New York: Academic
Press.
Hannan, E.J. (1970) Multiple Time Series. New York: Wiley.
Hannan, E.J. and M. Deistler (1988) Statistical Theory of Linear Systems. New York: Wiley.
* Hamilton, J.D. (1994) Time Series Analysis. Princeton: Princeton University Press.
Harvey, A.C. (1993) Time Series Models. Hemel Hempstead: Harvester Whaetsheaf.
Harvey, A.C. (1990) Forecasting Structual Time Series Models and the Kalman Filter. New York:
Cambridge University Press.
Hendry, D. F. (1995) Dynamic Econometrics. Oxford: Oxford University Press.
Hsaio, C. (2003) Analysis of Panel Data. (2’nd Ed.) Cambridge:: Cambridge University Press.
Hylleberg, S. (1992) Modelling Seasonality. Oxford: Oxford University Press.
* Lutkepohl, H. (1993) Introduction to Multiple Time Series Analysis, 2nd ed. New York: Springer
Verlag.
* Lutkepohl, H. (2005) A New Introduction to Multiple Time Series Analysis, New York: Springer
Verlag.
* Lutkepohl, H. and M. Kratzig (2004) Applied Time Series Econometricss, Cambridge University
Press.
Maddala, G. S. and I-M. Kim (1998). Unit Roots, Cointegration, and Structural Change. Cambridge
University Press.
* Martin, V., S. Hurn and D. Harris (2012). Econometric Modelling with Time Series: Specification,
Estimation and Testing. Cambridge University Press.
Matyas, L. (1999). Generalized Methods of Moments Estimation., Cambridge: Cambridge University
Press.
Mills, T. C. (1990) Time Series Techniques for Economists. Cambridge: Cambridge University
Press
Potscher B. and I. Prucha, “ Dynamic Nonlinear Econometric Models” New York: Springer.
Priestley, M.B. (1981) Spectral Analysis and Time Series. Vol. 1, New York: Academic Press.
Rao, B.B. (1994) Cointegration for the Applied Economist. St. Martin's Press.
5
Reinsel, G. (1993) Elements of Multivariate Time Series Analysis. New York: Springer.
Taniguchi, M. and Y. Kakizawa (2000). Asymptotic Theory of Statistical Inference for Time Series.
New York: Springer Verlag.
Tong, H. (1990) Non-Linear Time Series: A Dynamical System Approach. Oxford: Clarendon
Press.
Van de Vaart (1998). Asymptotic Statistics. Cambridge University Press.
Watson, M. (1995) "Vector Autoregressions and Cointegration." In R.F. Engle and D. McFadden,
eds., Handbook of Econometrics, Vol. 4. Amsterdam: North Holland.
West, M. and P.J. Harrison (1989) Bayesian Forecasting and Dynamic Models. New York: SpringerVerlag.
White, H. (1994) Estimation, Inference and specification Analysis. Cambridge: Cambridge University
Press.
White, H. (2002) Asymptotic Theory for Econometricians. (Revised Edition) San Diego: Academic
Press.
Whittle, P. (1984) Prediction and Regulation, 2nd ed. Oxford: Blackwell.
Wooldridge, J. M. (1995) "Estimation and Inference for Dependent Processes" in R. F. Engle and
D. L. McFadden Handbook of Econometrics Vol IV. Amsterdam: North Holland.
Yaglom, A.M. (1962) An Introduction to the Theory of Stationary Random Functions. New York:
Dover.
1. Ideas and Approaches
* Phillips P. C. B. (1989 & 1995) Lecture notes
Phillips, P.C.B. (1992, 2008) "Unit Roots." In P. Newman, M. Milgate and J. Eatwell, eds., The New
Palgrave Dictionary of Money and Finance, 726-730.
Phillips, P.C.B. (1995) "Unit Roots and Cointegration: Recent Books and Themes for the Future,"
Journal of Applied Econometrics
Phillips P. C. B. (1998) "Econometric Analysis of Nonstationary Data", IMF Lectures
Phillips, P. C. B. (1998). "New Tools for Understanding Spurious Regressions". Econometrica, 66,
1299-1326.
Phillips, P. C. B. (2001): "Descriptive Econometrics for Nonstationary Time Series with Empirical
Illustrations," Journal of Applied Econometrics, 16, 389-413.
6
Phillips P. C. B. (2003) “Laws and Limits of Econometrics”, Economic Journal, Vol. 113, No. 486,
March, 2003, pp. C26-C52.
Phillips, P. C. B. (2005) “Challenges of Trending Time Series Econometrics” Mathematics and
Computers in Simulation, 68, 401-416.
Phillips, P. C. B. (2005) “Automated Discovery in Econometrics” Econometric Theory, 21, 3-20.
Phillips, P. C. B. (2009) “Econometric Theory and Practice,” Econometric Theory, 25, 583-586.
2. Karhunen Loève Representation, Unit Roots and Cointegration, Optimal Estimation
Johansen, S. (1995), Likelihood-based inference in cointegrated vector autoregressive models, Oxford:
Oxford University Press.
Phillips, P. C. B. (1986). "Understanding spurious regressions in econometrics," Journal of
Econometrics 33, 311--340.
Phillips, P. C. B. (1987). "Time Series Regression with a Unit Root," Econometrica, 55, 277--302.
Phillips, P. C. B. (1991b). "Optimal inference in cointegrated systems," Econometrica 59, 283--306.
Phillips, P. C. B. (1998). "New Tools for Understanding Spurious Regressions". Econometrica, 66,
1299-1326.
Phillips, P. C. B (2001). "New Unit Root Asymptotics in the Presence of Deterministic Trends".
Journal of Econometrics, 11, 323-353.
Phillips, P. C. B. (2013). “Optimal Estimation of Cointegrated Systems with Irrelevant Instruments”,
Journal of Econometrics (forthcoming)
Phillips, P. C. B. and B. E. Hansen (1990). "Statistical inference in instrumental variables regression
with I(1) processes," Review of Economic Studies, 57, 99--125.
Phillips, P. C. B. and M. Loretan (1991). "Estimating long-run economic equilibria," Review of
Economic Studies 59, 407--436.
Saikkonen, P. (1991). "Asymptotically efficient estimation of cointegration regressions," Econometric
Theory 7, 1--21.
Stock, J. H. and M. W. Watson (1993). "A simple estimator of cointegrating vectors in higher order
integrated systems," Econometrica 61, 783--820.
3. Classical and Bayesian Asymptotics for time series and Model Selection
Chen, C. F. (1985). ``On asymptotic normality of limiting density functions with Bayesian
implications,''Journal of the Royal Statistical Society, Series B, 47, 540--546.
Hartigan, J. A. (1983). Bayes Theory. New York: Springer-Verlag.
7
Heyde, C. C. and I. M. Johnstone (1979). ``On asymptotic posterior normality for stochastic processes,''
Journal of the Royal Statistical Society, 41, 184--189.
Kim, J. Y. (1994). "Bayesian asymptotic theory in a time series model with a possible nonstationary
process," Econometric Theory, 10, 764--773.
Kim J. Y. (1998) "Large Sample Properties of Posterior Densities Bayesian Information Crterion and
the Likelihood Principle in Nonstationary Time Series Models," Econometrica, 66, 359-380.
Le Cam, L. and G. L. Yang (1990). Asymptotics in Statistics: Some Basic Concepts. New York:
Springer
* Phillips, P.C.B. (1996) "Econometric Model Determination " Econometrica, 64, 763-812.
* Phillips, P. C. B. and W. Ploberger (1996). ``An asymptotic theory of Bayesian inference for time
series,'' Econometrica, 64, 381-413.
Ploberger W. and P. C. B. Phillips (2003) "Empirical Limits for Time Series Econometric Models",
Econometrica, Vol. 71, No. 2, pp. 627-673.
* Schwarz, G. (1978) "Estimating the dimension of a model," Annals of Statistics, 6:461-464.
Sweeting, T. J. and A. O. Adekola (1987). ``Asymptotic posterior normality for stochastic processes
revisited,'' Journal of the Royal Statistical Society, Series B, 49, 215--222.
4. Strict Stationarity and Ergodic Theory
Cramer, H. and M.R. Leadbetter (1967) Stationary and Related Stochastic Processes. New York:
Wiley.
* Dhrymes (1989) op. cit.
Rozanov, Y.A. (1967) Stationary Random Processes. San Francisco: Holden Day.
* Stout, W.F. (1974) Almost Sure Convergence. New York: Academic Press.
Walters, P. (1982) An Introduction to Ergodic Theory. New York: Springer.
5. Projections and the Wold Decomposition
Anderson (1971) op. cit.
* Brockwell and Davis (1993) op. cit.
* Hannan (1970) op. cit.
Whittle (2002) op. cit.
8
6. Weak Dependence and Mixing Processes
* Davidson J. (1995) op. cit.
Gallant A. R. and H. White (1988) A Unified Theory of Estimation and Inference for Nonlinear
Dynamic Models. New York: Basil Blackwell.
Ibragimov, I.A. and Y.V. Linnik (1971) Independent and Stationary Sequences of Random Variables.
Groningen: Wolters-Noordhoff.
Potscher B. and I. Prucha (1997) op. cit.
* White, H. (2002) op. cit.
White, H. and I. Domowitz (1984)
Econometrica, 52:143-162.
"Nonlinear Regression with Dependent Observations,"
7. BN Decomposition and Phillips-Solo Device
* Beveridge, S. and C. R. Nelson (1981). "A new approach to decomposition of economic time series
into permanent and transitory components with particular attention to measurement of the
`business cycle','' Journal of Monetary Economics, 7, 151--174.
* Phillips, P.C.B. and V. Solo (1992) "Asymptotics for Linear Processes," Annals of Statistics, 20:9711001.
8. Martingales, Martingale Convergence Theory and Strong Laws for Dependent Sequences
Billingsley, P. (1979) Probability and Measure. New York: Wiley.
Doob, J.L. (1953) Stochastic Processes. New York: Wiley.
* Hall, P. and C.C. Heyde (1980) Martingale Limit Theory and its Application. New York: Academic
Press.
McLeish, D.L. (1975) "A Maximal Inequality and Dependent Strong Laws," Annals of Probability,
3:829-839.
* Phillips, P.C.B. and V. Solo (1992) op. cit.
9. Central Limit Theory for Dependent Variables
Davidson J. (1995) op. cit.
* Hall and Heyde (1980) op. cit.
* Phillips and Solo (1992) op. cit.
White, H. (2002) op. cit.
9
10. Spectrum, HAC and Long Run Variance Matrix Estimation
* Andrews, D.W.K. (1991) "Heteroskedasticity and Autocorrelation Consistent Covariance Matrix
Estimation," Econometrica, 817-858.
Andrews, D.W.K. and J.C. Monahan (1992) "An Improved Heteroskedasticity and autocorrelation
Consistent Covariance Matrix Estimator," Econometrica, 60, 953-966.
Den Haan, W.J., and A. Levin, 1997, "A practitioner's guide to robust covariance matrix estimation,"
in Handbook of Statistics 15, G.S. Maddala and C.R. Rao, eds., Elsevier (Amsterdam), pp.299342.
Den Haan, W.J., and A. Levin, 2000, "Robust covariance matrix estimation with data-dependent
prewhitening order", Working Paper 2000-11, University of California, San Diego
* Hannan, E. J. (1970) op. cit.
Lee, C. C. and P. C. B. Phillips (1994) "An ARMA-prewhitened long run variance estimator",
Yale University, mimeographed.
Newey, W.K. and K.D. West (1987) "A Simple Positive Semi-Definite, Heteroskedasticity and
Autocorrelation Consistent Covariance Matrix," Econometrica, 55, 703-708.
Parzen, E. (1957) "On Consistent Estimates of the Spectrum of a Stationary Time Series," Annals of
Mathematical Statistics, 28:329-348.
Phillips, P. C. B. (2004). "HAC Estimation by Automated Regression." Econometric Theory, 21, 116142.
Phillips, P. C. B., Y. Sun and S. Jin (2006) “Spectral Density Estimation and Robust Hypothesis
Testing using Steep Origin Kernels without Truncation”, International Economic Review.
Phillips, P. C. B., Y. Sun and S. Jin (2006) “Long Run Variance Estimation and Robust Regression
Testing using Sharp Origin Kernels with No Truncation” (with Yixiao Sun and Sainan Jin),
Journal of Statistical Planning and Inference, 2006 (forthcoming)
* Priestley (1981) op. cit.
Robinson, P.M. (1998), “Inference-without-smoothing in the Presence of Nonparametric
Autocorrelation,” Econometrica ,66, 1163-1182.
Sul, D., C-Y Choi and P. C. B. Phillips (2005) “Prewhitening Bias in HAC Estimation”, Oxford
Bulletin of Economics and Statistics, 67, 517-546.
Sun, Y., P. C. B. Phillips, and S. Jin (2008) “Optimal Bandwidth Selection in HeteroskedasticityAutocorrelation Robust Testing” Econometrica. 76, 175-194.
White, H. (1980) "A Heteroskedasticity-Consistent Covariance Matrix Estimator and a Direct Test of
Heteroskedasticity," Econometrica, 48, 817-838.
10
White, H. (2002) op. cit.
11. Spectral Regression Theory
Corbae, D., S. Ouliaris and P. C. B. Phillips (2002) "Band Spectral Regression with Trending Data".
Econometrica, 70, 1067-1110.
* Hannan, E. J. (1963) "Regression for Time Series" in M. Rosenblatt (Ed.) Time Series Analysis,
New York: Wiley.
* Hannan (1970) op. cit.
Phillips, P. C. B. (1997) “New developments on Hannan Regression”, Ted Hannan Lecture,
Australasian meetings of Econometric Society, Melbourne.
Robinson, P.M. (1991) "Automatic frequency domain inference on semiparametric and nonparametric
models," Econometrica,59, 1329-1364.
Xiao, Z. and P. C. B. Phillips, (1998). “Higher Order Approximations for Frequency Domain Time
Series Regression”, Journal of Econometrics, Vol. 86, 1998, pp. 297-336
12. VAR'S, BVAR's, Impulse Response Analysis
Cooley, T.B. and S.F. LeRoy (1985) "Atheoretical Macroeconometrics: A Critique," Journal of
Monetary Economics, 16:283-308.
* Hamilton (1994) Chs. 11, 12.
Litterman, R.B. (1986) "Forecasting with Bayesian Vector Autoregressions:
Experience," Journal of Business and Economic Statistics, 4:25-38.
Five Years of
Litterman, R.B. and L. Weiss (1985) "Money, Real Interest Rates, and Output: A Reinterpretation of
Postwar U.S. Data," Econometrica, 53:129-156.
* Lutkepohl, H. (1990) "Asymptotic Distributions of Impulse Response Functions and Forecast Error
Variane Decompositions of Vector Autoregressive Models," Review of Economics and Statistics,
72:116-125.
* Lutkepohl, H. (1993) op.cit., Ch. 5.
Phillips, P.C.B. (1995a) "Bayesian Model Selection and Prediction with Empirical Applications,"
Journal of Econometrics, 69, 289-332
Phillips, P.C.B. (1995b) "Bayesian Prediction: A Response," Journal of Econometrics,.69, 351-365.
* Phillips, P.C.B. (1998) "Impulse response and forecast error asymptotics in nonstationary VAR's."
Journal of Econometrics, 83, 21-56.
Runkle, D. (1987) "Vector Autoregressions and Reality," Journal of Business and Economic Statistics,
5(4):437-432.
11
* Sims, C.A. (1980) "Macroeconomics and Reality," Econometrica, 48:1-48.
Todd, R.M. (1990) "Vector Autoregression Evidence on Monetarism: Another Look at the
Robustness Debate," Federal Reserve Bank of Minneapolis Quarterly Review, 19-37.
Todd, R.M. (1995) "Improving Economic Forecasting with Bayesian Vector Autoregression," Federal
Reserve Bank of Minneapolis Quarterly Review, 4:18-29.
Zellner, A. and C.K. Min (1992) "Bayesian Analysis, Model Selection and Prediction," University of
Chicago, Mimeographed.
13. Long Memory Models and Econometric Methods
* Baillie, R. T. (1996). "Long memory processes and fractional integration in econometrics". Journal of
Econometrics, 73, 5-59.
Baillie, R. T. and T. Bollerslev (1994). “Long memory in the forward premium”. Journal of
International Money and Finance, 13, 565-571.
Geweke J. and S. Porter-Hudak (1983) "The estimation and application of long memorey time series
models”. Journal of Time Series Analysis, 4, 221-237.
Granger, C. W. J. (1980). “Long memory relationships and the aggregation of dynamic models”.
Journal of Econometrics, 14, 227-238.
* Granger, C. W. J. and R. Joyeux (1980). “An introduction to long memory time series models and
fractional differencing”. Journal of Time Series Analysis, 1, 15-39.
* Hosking, J. R. M. (1981). “Fractional differencing”. Biometrika, 68, 165-176.
Kunsch, H. (1986). “Discrimination between monotonic trends and long-range dependence”. Journal
of Applied Probability, 23, 1025-1030.
Mandelbrot, B. B. and J. W. Van Ness (1968). “Fractional Brownian motions, fractional Brownian
noises and applications”. SIAM Review, 10, 422-437.
Mandelbrot, B. B. and J. Wallis (1968). “Noah, Joseph and operational hydrology”. Water Resources
Research, 4, 909-918.
* Phillips, P. C. B. (1999) "Discrete Fourier Transforms of Fractional Processes”. Cowles Foundation
Discussion Paper #1243, Yale University.
Phillips, P. C. B. & K. Shimotsu (2004) “Local Whittle Estimation in Nonstationary and Unit Root
Cases,” Annals of Statistics, 32, 656-692.
Phillips, P. C. B. (2006) “Unit Root Log Periodogram Regression”, Journal of Econometrics,
* Robinson, P. M. (1995) "Log periodogram regression of time series with long range dependence”.
Annals of Statistics, 23, 1048-1072.
12
Robinson, P. M. (1995) "Gaussian semiparametric estimation of time series with long range
dependence”. Annals of Statistics, 23, 1630-1661.
Shimotsu, K. & Phillips, P. C. B (2005) “Exact Local Whittle Estimation of Fractional Integration”
Annals of Statistics, 33, 1890-1933
Shimotsu, K. & Phillips, P. C. B (2006) “Local Whittle Estimation of Fractional Integration and Some
of its Variants”, Journal of Econometrics.
Sowell, F. B. (1986). “Fractionally integrated vector time series”. Ph.D. dissertation (Duke University,
Durham, NC).
Sowell, F. B. (1992). “Maximum likelihood estimation of stationary univariate fractionally integrated
time series models”. Journal of Econometrics, 53, 165-188.
Sun, Y. and P. C. B. Phillips (2003). "Nonlinear Log-Periodogram Regression for Perturbed Fractional
Processes", Journal of Econometrics, Vol. 115, No. 2, pp. 355-389.
Journal of Econometrics, Vol. 73 (1996) [special issue].